The unfolded equation has 5 singular points. The points \(x=\sqrt{\varepsilon }, x=-\sqrt{\varepsilon },x=1/\sqrt{\varepsilon }\)  and \(x=-\sqrt{\varepsilon }\)  are regular singularities.
The transformation \(x=1/t\) 
changes the unfolded equation into the equation
\(\ddot{w} &+&\Big [\frac{2-\alpha }{t} -\frac{\gamma }{2 \sqrt{\varepsilon }}\left(\frac{1}{t-\sqrt{\varepsilon }} - \frac{1}{t+\sqrt{\varepsilon }}\right)-\frac{\beta }{2 \sqrt{\varepsilon }} \left(\frac{1}{\frac{1}{\sqrt{\varepsilon }} -t}+ \frac{1}{\frac{1}{\sqrt{\varepsilon }}+t}\right) \\[0.25ex]&-&\frac{\alpha }{2} \left(\frac{1}{\frac{1}{\sqrt{\varepsilon }}-t} -\frac{1}{\frac{1}{\sqrt{\varepsilon }}+t}\right)\Big ]\,\dot{w} \\[0.2ex]&+&\left[\frac{\delta \,\varepsilon }{t} - \frac{q}{t^2} + \frac{\delta }{t^3}+\frac{\delta \,\varepsilon }{2}\left(\frac{1}{\frac{1}{\sqrt{\varepsilon }}-t} - \frac{1}{\frac{1}{\sqrt{\varepsilon }}+t}\right)-\frac{q\,\sqrt{\varepsilon }}{2}\left(\frac{1}{\frac{1}{\sqrt{\varepsilon }}-t} +\frac{1}{\frac{1}{\sqrt{\varepsilon }}+t}\right) \right]\,w=0\,.\) 
The point \(t=0\)  is a regular singularity for the last equation if and only if \(\delta =q=0\) . Hence the unfolded equation
is a Fuchsian equation if and only if \(\delta =q=0\) .
This ends the proof.
\(\Box \)
Theorem REF  is the main motivation for giving our attention to the unfolding of the reducible DCHE
\(w^{\prime \prime } + \left[\frac{\alpha }{x} - \frac{\beta }{x^2} - \gamma \right]\,w^{\prime }=0\,,\) 
which is obtained from the equation (REF ) after the transformation \(x \rightarrow -x\) .
Since the rotation \(x \rightarrow e^{i \delta }\,x,\,\,\delta =\arg (\beta )\)  changes the equation (REF ) into the
equation
\(\,w^{\prime \prime } + \left[\frac{\alpha }{x} - \frac{|\beta |}{x^2} - \gamma \,e^{i \delta }\right]\,w^{\prime }=0\,\) 
without loss of generality throughout this paper we assume that \(\beta \)  is a real non-negative parameter.
The corresponding
unfolded Fuchsian equation writes
\(w^{\prime \prime } &+&\Big [\frac{\alpha }{2}\left(\frac{1}{x-\sqrt{\varepsilon }}+ \frac{1}{x+\sqrt{\varepsilon }}\right) -\frac{\beta }{2 \sqrt{\varepsilon }}\left(\frac{1}{x-\sqrt{\varepsilon }} - \frac{1}{x+\sqrt{\varepsilon }}\right) \\[0.25ex]&-&\frac{\gamma }{2 \sqrt{\varepsilon }}\left(\frac{1}{\frac{1}{\sqrt{\varepsilon }}-x} +\frac{1}{\frac{1}{\sqrt{\varepsilon }}+x}\right)\Big ]\,w^{\prime }=0\,.\nonumber \) 
We denote the singular points of the unfolded equation by
\(x_L=-\sqrt{\varepsilon }, x_R=\sqrt{\varepsilon }, x_{LL}=-1/\sqrt{\varepsilon }, x_{RR}=1/\sqrt{\varepsilon }\) 
and \(x_{\infty }=\infty \) .
Obviously the singular points \(x_L\)  and \(x_R\)  are obtained by the unfolding of \(x=0\)  while
\(x_{LL}\)  and \(x_{RR}\)  are obtained by the unfolding of \(x=\infty \)  of (REF ). It is expected that \(x_{\infty }\)  is
also a result of the unfolding of \(x=\infty \) . In this paper comparing the analytic invariants of both equations
we confirm this conjecture. More precisely, we will show that the analytic invariants of the DCHE around the origin
are realizable as a limit when \(\sqrt{\varepsilon } \rightarrow 0\)  of the analytic invariants of the unfolded equation only
around resonant singularities \(x_L\)  and \(x_R\) . This phenomenon implies that the monodromy around \(x_{LL}, x_{RR}\) 
and \(x_{\infty }\)  is responsible for the unfolding of the analytic invariants around the singularity \(x=\infty \) 
of the DCHE. The study of the nature of the unfolding of \(x=\infty \)  is left to another project.
Similar kind of problems related to the unfolding and confluence of singularities of the differential equations
have been studied in the works of Bolibrukh [1]}, Glutsyuk [2]}, [3]}, [4]}, Hurtubise, Lambert and Rousseau
[5]}, [6]}, [7]}, [8]}, Klimeš [9]}, [10]}, Ramis [11]}, Stoyanova [12]}, [13]}, Zhang Z.
In the works of Buchstaber and Glutsyuk [14]}, El-Jaick and Figueiredo [15]}, Roseau [16]}, Tertychniy [17]}
have been studied solutions
space and Stokes phenomenon of the families
of double confluent Heun equations.
The kind of singularity at the origin depends on the parameter \(\beta \) . When \(\beta =0\)  the origin is a regular
singular point and the DCHE (REF ) degenerates into a Bessel type of equation. We introduce the notion
of unfolded monodrmy (see Definition REF ) as an analog of the unfolded Stokes matrix introduced by
Lambert and Rousseay in [6]}. The unfolded monodromy measures geometrically the transformation
of the monodromy around the regular singularity at the origin after a general symmetric unfolding.
The reducibility allows us to prove in Section 3.3, Theorem REF 
that when \(\beta =0\) 
the monodromy around the origin of the equation (REF ) is realizable as a limit when \(\sqrt{\varepsilon } \rightarrow 0\)  of the
unfolded monodromy which depends analytically on \(\sqrt{\varepsilon }\) .
The main result in Section 3 states
that when the origin is a resonant singularity the unfolded monodromy matrix is expressed as a product of the local monodromy
matrices of the unfolded equation around resonant singular points \(x_L\)  and \(x_R\)  (see Theorem REF ).
In Section 3.2, Lemma REF  we demonstrate by a direct computation that the coefficients in the logarithmic terms
of the solution of the unfolded equation have limits when \(\sqrt{\varepsilon } \rightarrow 0\)  and both of the limits are equal to
\(\infty \)  whose sign depends on the parameter \(\alpha \) . It turns out that the sum of these coefficients has a finite limit
when \(\sqrt{\varepsilon } \rightarrow 0\)  which is equal to the monodromy around the origin of the solution of the DCHE
(see Corollary REF ). Lemma REF  together with Lemma REF  in Section 4.1 fix the main
difference between the unfolding of a regular singularity and an irregular singularity. In Lemma REF  we show
explicitly that when the origin is an irregular singularity the coefficient in the logarithmic term of the
solution of the unfolded equation has a finite limit when \(\sqrt{\varepsilon } \rightarrow 0\) . Moreover, this limit
multiplied by \(2\,\pi \,i\)  is equal to the corresponding Stokes multiplier.
In [13]} we have shown by a direct computation that when \(\alpha =2, \beta \ne 0\)  the Stokes matrices at \(x=0\)  and \(x=\infty \) 
of the reducible double confluent Heun equation (REF ) are realizable as a limit of the part of the monodromy matrices
around a resonant singularity of the general reducible Heun equation (REF ).
In Section 4.1 based on the recent works of Lambert, Rousseau, Hurtubise and Klimeš [5]}, [10]}, [6]}, [7]}
we extend the result in [13]} to an arbitrary reducible DCHE (REF ) without studying this equation.
In fact this theoretical result allows us to derive the Stokes multiplier at the origin from the unfolded equation.
In Section 4.2 we build explicit fundamental matrix solution at the origin with respect to which the
Stokes multiplier is equal to that one obtained in Section 4.1. It turns out that the reducible DCHE (REF )
admits a solution which is holomorphic in whole \({\mathbb {C}}^*\)  if and only if
the parameters \(\alpha , \beta \)  and
\(\gamma \)  satisfy either the relation
\(\sum _{k=0}^{\infty } \frac{(-1)^k\,\beta ^k\,\gamma ^k}{k!\,\Gamma (2-\alpha +k)}=0,\quad \alpha \notin {\mathbb {N}}\,,\) 
or
\(\sum _{k=0}^{\infty } \frac{(-1)^k\,\beta ^k\,\gamma ^k}{k!\,\Gamma (\alpha +k)}=0,\quad \alpha \in {\mathbb {N}}\,,\) 
where \(\Gamma (z)\)  is the Euler Gamma function.
The relations (REF ) and (REF ) associate the parameters \(\alpha , \beta \)  and \(\gamma \)  with the Bessel function
\(\,J_{\alpha }(x)=\left(\frac{x}{2}\right)^{\alpha }\,\sum _{n=0}^{\infty }\frac{(-1)^n\,(x/2)^{2 n}}{n!\,\Gamma (n+\alpha +1)}\,\) 
of the first kind of order \(\alpha \) .
This paper is organized as follows. In Section 3 we study the unfolding of the regular singularity at the
origin and the corresponding monodromy. In Section 4 we deal with the unfolding of the irregular singularity
at the origin and the corresponding Stokes phenomenon. The paper contains also an Appendix where we
confirm Corollary REF  by a direct computation for lower values of the parameter \(\alpha \) .
Since this paper appears as an extension of [13]} we use without any effort some definitions and facts
from [13]}.
Global solutions and logarithms, singular direction
Theorem 2.1  
The equation (REF ) possesses a fundamental set of solutions \(\lbrace w_1(x, 0), w_2(x, 0)\rbrace \)  of the form
\(w_1(x, 0)=1,\qquad w_2(x, 0)=\int _{\Gamma (x, 0)}z^{-\alpha }\,e^{-\frac{\beta }{z}}\,e^{\gamma \,z}\, d z\,.\) 
The path of integration \(\Gamma (x, 0)\)  is a path in a direction taken in such a way that the function
\(w_2(x, 0)\)  is a solution of equation (REF ).

We have a similar result for the equation (REF ).
Theorem 2.2  
The equation (REF ) possesses a fundamental set of solution \(\lbrace w_1(x, \varepsilon ), w_2(x, \varepsilon )\rbrace \) 
of the form
\(& &w_1(x, \varepsilon )=1,\\[0.1ex]& &w_2(x, \varepsilon )=\int _{\Gamma (x, \varepsilon )}(z-\sqrt{\varepsilon })^{\frac{\beta }{2 \sqrt{\varepsilon }}-\frac{\alpha }{2}}\,(z+\sqrt{\varepsilon })^{-\frac{\beta }{2 \sqrt{\varepsilon }}-\frac{\alpha }{2}}\,\left(\frac{\frac{1}{\sqrt{\varepsilon }}+z}{\frac{1}{\sqrt{\varepsilon }}-z}\right)^{\frac{\gamma }{2 \sqrt{\varepsilon }}}\,d z\,,\nonumber \) 
which depends analytically on \(\sqrt{\varepsilon }\) .
The path of integration \(\Gamma (x, \varepsilon )\)  such that \(\Gamma (x, \varepsilon ) \rightarrow \Gamma (x, 0)\)  when
\(\sqrt{\varepsilon } \rightarrow 0\)  is a path in the same direction and with the same base point \(x\)  as the path \(\Gamma (x, 0)\)  from Theorem REF  and taken in such a way
that the function \(w_2(x, \varepsilon )\)  is a solution of the equation (REF ).

As a direct consequence of Theorem REF  and Theorem REF  we construct fundamental matrices of equations (REF ) and
(REF ).
Theorem 2.3  
The equations (REF ) and (REF ) possess a fundamental matrix solution \(\Phi (x, \cdot )\)  in the form
\(\Phi (x, \cdot )=\left(\begin{array}{cc}1 & w_2(x, \cdot )\\0 & w^{\prime }_2(x, \cdot )\end{array}\right), \quad \cdot =\lbrace 0, \varepsilon \rbrace \,.\) 

In this paper we study both equations in the
punctured disk \(D_R\)  around the origin with a finite small radius \(R\)
\(\,D_R:=\lbrace x\in {\mathbb {C}}\,|\, 0 < |x| < R\rbrace \,.\,\) 
The radius \(R\)  is so chosen that the points \(x_L\)  and \(x_R\)  belong to \(D_R\)  while the points \(x_{LL}\)  and \(x_{RR}\) 
do not belong to \(D_R\) . Then
Proposition 2.4  
When \(\sqrt{\varepsilon } \rightarrow 0\)  the fundamental set of solutions \(\lbrace w_1(x, \varepsilon ), w_2(x, \varepsilon )\rbrace \) 
of the unfolded equation fixed by Theorem REF  converges uniformly on compact sets of \(D_R\)  to the fundamental set of solutions
\(\lbrace w_1(x, 0), w_2(x, 0)\rbrace \)  of the DCHE fixed by Theorem REF .

Let us determine when the solution \(w_2(x, \varepsilon )\)  of the unfolded equation can contain logarithmic terms near
the singular points \(x_j, j=L, R\) . Recall that form the local theory of the Fuchsian singularity such a singular point
is called a resonant singularity. When \(\beta =0\)  the points \(x_L\)  and \(x_R\)  are together
either non-resonant or resonant singularities for the unfolded equation. In particular, they both are resonant
singularities if and only if \(\alpha \in 2 {\mathbb {N}}\) . In the next section we consider the equations (REF ) and
(REF ) under the restriction
\(\beta =0,\qquad \alpha \in 2 {\mathbb {N}}\,.\) 
Note that under the restriction (REF ) the origin is a resonant regular singularity too.
When \(\beta > 0\)  we choose the path \(\Gamma (x, 0)\)  in (REF ) to be a path from 0 to \(x\)  approaching 0 in the direction
\({\mathbb {R}}^{+}\) . Then
the corresponding unfolded path \(\Gamma (x, \varepsilon )\)  is a path taken
in the same direction \({\mathbb {R}}^{+}\)  from \(\sqrt{\varepsilon }\)  to the same base point \(x\) . This choice of the path
\(\Gamma (x, \varepsilon )\)  implies that \(\varepsilon \)  is a real positive parameter of unfolding and that \(x_L\)  will be the resonant singularity.
In particular in Section 4 we consider the unfolded equation under the restriction
\(\frac{\beta }{2 \sqrt{\varepsilon }} + \frac{\alpha }{2}\in {\mathbb {N}},\quad \frac{-\beta }{2 \sqrt{\varepsilon }} + \frac{\alpha }{2}\notin {\mathbb {N}}\,.\) 
From Definition 6.15 in [13]} it follows that \(\theta =\arg (0-\beta )=\arg (-\beta )=\pi \)  is the only possible
singular direction at the origin of the DCHE.

The unfolding of the monodromy around the origin
In this section we deal with the equations (REF ) and (REF ) when the parameters \(\alpha \)  and \(\beta \)  satisfy the
condition (REF ).
Using the rotation \(x \rightarrow x\,e^{i \delta }\)  where \(\delta =\arg (\sqrt{\varepsilon })\)  we always can fix \(\sqrt{\varepsilon }\) 
to be a real and positive. Due to this property in this section we assume that the paths \(\Gamma (x, 0)\)  and \(\Gamma (x, \varepsilon )\) 
in (REF ) and (REF ) are paths taken in \({\mathbb {R}}^{+}\) . The path \(\Gamma (x, 0)\)  is from
1 to \(x\)  while the path \(\Gamma (x, \varepsilon )\)  is a path
from \(1+\sqrt{\varepsilon }\)  to the same \(x\)  where \(x\in D_R\) . We denote by \(\Phi _0(x, 0)\)  and \(\Phi _0(x, \varepsilon )\)  the fundamental matrix solutions from (REF ) corresponding to the so chosen paths \(\Gamma (x, 0)\)  and \(\Gamma (x, \varepsilon )\) .
The monodromy around the origin of the DCHE 
Since the origin is a regular point for the DCHE its unfolding causes an unfolding of the monodromy around it.
To compute this monodromy we rewrite the fundamental matrix \(\Phi _0(x, 0)\)  in an appropriate form.
Directly from (REF ) and (REF ) we have
Theorem 3.1  
Assume that the condition (REF ) holds. Then the fundamental matrix solution \(\Phi _0(x, 0)\)  of the equation
(REF ) is represented in a neighborhood of the origin as
\(\Phi _0(x, 0)=\exp (G x)\,\,H(x)\,x^{\Lambda }\,x^J\,,\) 
where
\(\,G= {\mathrm {diag}} (0, \gamma ),\quad \Lambda = {\mathrm {diag}} (0, -\alpha )\,.\,\) 
The matrix \(H(x)\)  is defined as
\(\,H(x)=\left(\begin{array}{cc}1 & x\,\varphi (x)\\0 & 1\end{array}\right)\,,\,\) 
where \(\varphi (x)\)  is a holomorphic function in a neighborhood of the origin.
The matrix \(J\)  is given by
\(\,J=\left(\begin{array}{cc}0 &\lambda \\0 &0\end{array}\right)\,,\,\) 
where
\(\lambda =\frac{\gamma ^{\alpha -1}}{(\alpha -1)!}\,.\) 

The monodromy of the fundamental matrix solution \(\Phi _0(x, 0)\)  around the origin is described
by the local monodromy matrix \(M_0\in GL_2({\mathbb {C}})\)
\(M_0=e^{2 \pi \,i\,\Lambda }\,e^{2 \pi \,i\,J}=e^{2 \pi \,i\,J}=\left(\begin{array}{cc}1 & 2 \pi \,i\,\lambda \\0 & 1\end{array}\right)\,,\) 
where \(\lambda \)  is itroduced by (REF ).

The monodromy around \(x_L\)  and \(x_R\)  of the unfolded equation
In this section we compute the local monodromy matrices of the equation (REF ) under the restriction (REF ).
In the next theorem we describe the local behavior of the fundamental matrix \(\Phi _0(x, \varepsilon )\)  near the singular points
\(x_R\)  and \(x_L\)  when both of them are resonant singularities.
Theorem 3.2  
Assume that the condition (REF ) holds.
Then the fundamental matrix solution \(\Phi _0(x, \varepsilon )\)  of
the unfolded
equation depends analytically on \(\sqrt{\varepsilon }\)  and it is represented in a neighborhood of the origin which contains only the singular points
\(x_L\)  and \(x_R\)  as
\(\Phi _0(x, \varepsilon ) =G(x, \varepsilon )\,H(x, \varepsilon )\,(x-x_L)^{\frac{1}{2} \Lambda }\,(x-x_R)^{\frac{1}{2} \Lambda }\,(x-x_L)^{J_L(\varepsilon )}\, (x-x_R)^{J_R(\varepsilon )}\,,\) 
where
\(\,G(x, \varepsilon )=(x-x_{LL})^{-\frac{x_{LL}}{2}\,G}\, (x_{RR}-x)^{-\frac{x_{RR}}{2}\,G}\,.\,\) 
The matrix \(H(x, \varepsilon )\)  is a holomorphic matrix-function at the both singular
points \(x_L\)  and \(x_R\)  such that \(H(x_k, \varepsilon )=I_2, k=L, R\) . The matrices \(G\)  and \(\Lambda \)  are introduced in
Theorem REF . The matrix \(J_k(\varepsilon ), k=L, R\)  is given by
\(J_k(\varepsilon )=\left(\begin{array}{cc}0 &q_k\\0 &0\end{array}\right)\,,\) 
where the number \(q_k\)  is defined as
\(q_k= {\mathrm {Res}} \left( w_2^{\prime }(x, \varepsilon ),\,x=x_k\right),\quad k=L, R\,.\) 

Let us present the fundamental matrix \(\Phi _0(x, \varepsilon )\) 
in the form
\(& &\Phi _0(x, \varepsilon )=\left(\begin{array}{cc}1 & 0\\0 & w_2^{\prime }(x, \varepsilon )\end{array}\right)\left(\begin{array}{cc}1 &\int _{1+\sqrt{\varepsilon }}^x w^{\prime }_2(z, \varepsilon )\,d z\\0 &1\end{array}\right)\\[0.2ex]&=&G(x, \varepsilon )\,\left[(x-x_L) (x-x_R)\right]^{\frac{1}{2} \Lambda }\left(\begin{array}{cc}1 &\int _{1+\sqrt{\varepsilon }}^x w^{\prime }_2(z, \varepsilon )\,d z\\0 &1\end{array}\right)\,, \) 
where
\(\,G(x, \varepsilon )=(x-x_{LL})^{-\frac{x_{LL}}{2}\,G}\, (x_{RR}-x)^{-\frac{x_{RR}}{2}\,G}\,.\,\) 
Consider the function
\( w_2 (x, \varepsilon )\) . Since when \(\alpha \in 2 {\mathbb {N}}\)  the function
\(\frac{1}{[(z-\sqrt{\varepsilon })(z+\sqrt{\varepsilon })]^{\alpha /2}}\)  is a rational function it can be splited into
a finite sum in \(\alpha \)  number simpler ratios \(\frac{c_j}{(z-\sqrt{\varepsilon })^j}\)  and
\(\frac{d_j}{(z+\sqrt{\varepsilon })^j},\,1 \le j \le \alpha /2\)  where the coefficients \(c_j, d_j\)  are uniquely
determined. Then the function \(w_2(x, \varepsilon )\)  can be written
as
\(& &\int _{1+\sqrt{\varepsilon }}^x w^{\prime }_2(z, \varepsilon )\,d z=\int _{1+\sqrt{\varepsilon }}^x \frac{P(z, \varepsilon )}{(z -\sqrt{\varepsilon })^{\frac{\alpha }{2}}}\left(\frac{\frac{1}{\sqrt{\varepsilon }}-z}{\frac{1}{\sqrt{\varepsilon }}+z}\right)^{\frac{\gamma }{2 \sqrt{\varepsilon }}}d z\\[0.25ex]&+&\int _{1+\sqrt{\varepsilon }}^x \frac{Q(z, \varepsilon )}{(z +\sqrt{\varepsilon })^{\frac{\alpha }{2}}}\left(\frac{\frac{1}{\sqrt{\varepsilon }}+z}{\frac{1}{\sqrt{\varepsilon }}-z}\right)^{\frac{\gamma }{2 \sqrt{\varepsilon }}}d z\\[0.25ex]&=&q_R \log (x-x_R) + q_L \log (x-x_L) + (x-x_R)^{-\frac{\alpha }{2}+1}\,h(x-x_R) + (x-x_L)^{-\frac{\alpha }{2}+1}\,g(x-x_L)\,.\) 
Here \(P(z, \varepsilon )\)  and \(Q(z, \varepsilon )\)  are polynomials of degree at most \(\alpha /2-1\) .
The functions \(h(x-x_R)\)  and \(g(x-x_L)\)  are holomorphic functions at the both singular points \(x_R\)  and \(x_L\)  since the
function
\(\left(\frac{\frac{1}{\sqrt{\varepsilon }}+z}{\frac{1}{\sqrt{\varepsilon }} -z}\right)^{\frac{\gamma }{2 \sqrt{\varepsilon }}}\)  is a holomorphic function at the both singular
points \(x_j, j=L, R\) .
Then we can present
\(\Phi _0(x, \varepsilon )\)  as
\(& &G(x, \varepsilon )\left(\begin{array}{cc}1 &(x-x_R)(x-x_L)^{\frac{\alpha }{2}} h(x-x_R) + (x-x_L)(x-x_R)^{\frac{\alpha }{2}} g(x-x_L)\\0 &1\end{array}\right)\\[0.25ex]&\times &\left(\begin{array}{cc}1 &q_L \log (x-x_L) + q_R \log (x-x_R)\\0 &(x-x_L)^{-\frac{\alpha }{2}} (x-x_R)^{-\frac{\alpha }{2}}\end{array}\right)\\[0.25ex]&=&G(x, \varepsilon )\left(\begin{array}{cc}1 &(x-x_R) (x-x_L)^{\frac{\alpha }{2}} h(x-x_R) + (x-x_L) (x-x_R)^{\frac{\alpha }{2}} g(x-x_L)\\0 &1\end{array}\right)\\[0.25ex]&\times &[(x-x_L) (x-x_R)]^{\frac{1}{2} \Lambda } \left(\begin{array}{cc}1 &q_L \log (x-x_L) + q_R \log (x-x_R)\\0 &1\end{array}\right)\\[0.2ex]&=& G(x, \varepsilon )\left(\begin{array}{cc}1 &(x-x_R) (x-x_L)^{\frac{\alpha }{2}} h(x-x_R) + (x-x_L) (x-x_R)^{\frac{\alpha }{2}} g(x-x_L)\\0 &1\end{array}\right)\\[0.25ex]&\times &[(x-x_L) (x-x_R)]^{\frac{1}{2} \Lambda } (x-x_L)^{J_L(\varepsilon )} (x-x_R)^{J_R(\varepsilon )}=\Phi _0(x, \varepsilon )\,.\) 
This ends the proof.
\(\Box \)
Let \(x_0\in D_R \backslash {\mathbb {R}}\) . Let \(\gamma _L\)  and \(\gamma _R\)  be two closed loops, starting and ending at the
point \(x_0\) . The loop \(\gamma _L\)  (resp. \(\gamma _R\) ) encircles only the point \(x_L\)  (resp. \(x_R\) ) in the positive sense.
Thanks to Theorem REF  we can fix explicitly the corresponding local monodromy matrices corresponding to the loops
\(\gamma _L\)  and \(\gamma _R\) .
Theorem 3.3  
The local monodromy matrices \(M_k(\varepsilon ),\,k=R, L\)  of the perturbed equation
with respect to the fundamental matrix \(\Phi _0(x, \varepsilon )\) , introduced by
Theorem REF  are
given by
\(M_k(\varepsilon )=e^{2 \pi \,i\,J_k(\varepsilon )}, \quad k=L, R\,.\) 

Analytic continuation of the fundamental matrix \(\Phi _0(x, \varepsilon )\)  along a path \(\gamma _L\)  leads to the
new fundamental matrix \([\Phi _0(x, \varepsilon )]_{\gamma _L}\) . The connection between these two fundamental matrices
is measured geometrically by the monodromy matrix \(M_L(\varepsilon )\)
\(\,[\Phi _0(x, \varepsilon )]_{\gamma _L}=\Phi _0(x, \varepsilon )\,M_L(\varepsilon )\,.\,\) 
On the other hand thanks to the explicit local form of the matrix \(\Phi _0(x, \varepsilon )\)  from Theorem REF 
we find that
\([\Phi _0(x, \varepsilon )]_{\gamma _L}&=&G(x, \varepsilon )\, H(x, \varepsilon )\,(x-x_L)^{\frac{1}{2} \Lambda }\, e^{\pi \,i \Lambda }\\[0.2ex]&\times &(x-x_R)^{\frac{1}{2} \Lambda }\,(x-x_L)^{J_L(\varepsilon )} \,e^{2 \pi \,i\,J_L(\varepsilon )}\, (x-x_R)^{J_R(\varepsilon )}\\[0.2ex]&=&G(x, \varepsilon )\,H(x, \varepsilon )\,(x-x_L)^{\frac{1}{2} \Lambda }\\[0.2ex]&\times &(x-x_R)^{\frac{1}{2} \Lambda }(x-x_L)^{J_L(\varepsilon )} e^{2 \pi \,i\,J_L(\varepsilon )}(x-x_R)^{J_R(\varepsilon )} \,,\) 
since \(e^{\pi \,i \Lambda }=I_2\)  when \(\alpha \in 2 {\mathbb {N}}\) .
As a result we have
\([\Phi _0(x, \varepsilon )]_{\gamma _L}=\Phi _0(x, \varepsilon ) e^{2 \pi \,i\,J_L(\varepsilon )} \,,\) 
since the matrices \(e^{2 \pi \,i\,J_L(\varepsilon )}\)  and \((x-x_R)^{J_R(\varepsilon )}\)  commute.
In the same manner one can devide the formula for the monodromy matrix \(M_R(\varepsilon )\) .
This ends the proof.
\(\Box \)
In what follows we compute the numbers \(q_k, k=L, R\) .
Proposition 3.4  
Assume that the condition (REF ) holds.
Then the number \(q_R\)  is given by
\(q_R=\frac{1}{\left(\frac{\alpha -2}{2}\right)!}\sum _{k=0}^{\frac{\alpha -2}{2}}\left( \left(\begin{array}{c}\frac{\alpha -2}{2}\\k\end{array}\right)(-1)^{\frac{\alpha -2}{2}-k}\frac{\Gamma (\alpha -1-k)}{\Gamma (\frac{\alpha }{2})} (2 \sqrt{\varepsilon })^{-\alpha +1+k}\right)\,A_R\,,\) 
where
\(\,A_R=\left(\frac{\sqrt{\varepsilon }}{1+\varepsilon }\right)^k\,\left(\frac{1+\varepsilon }{1-\varepsilon }\right)^{\frac{\gamma }{2\sqrt{\varepsilon }}}\sum _{s=0}^k \left(\begin{array}{c}k\\s\end{array}\right)\frac{\Gamma (\frac{\gamma }{2 \sqrt{\varepsilon }} + s)\,\Gamma (\frac{\gamma }{2 \sqrt{\varepsilon }}+1)}{\Gamma (\frac{\gamma }{2 \sqrt{\varepsilon }})\,\Gamma (\frac{\gamma }{2 \sqrt{\varepsilon }}-k+s+1)}\left(\frac{1+\varepsilon }{1-\varepsilon }\right)^s\,.\,\) 
Similarly, the number \(q_L\)  is
given by
\(q_L=\frac{1}{\left(\frac{\alpha -2}{2}\right)!}\sum _{k=0}^{\frac{\alpha -2}{2}}\left( \left(\begin{array}{c}\frac{\alpha -2}{2}\\k\end{array}\right)(-1)^{\frac{\alpha -2}{2}-k}\frac{\Gamma (\alpha -1-k)}{\Gamma (\frac{\alpha }{2})} (-2 \sqrt{\varepsilon })^{-\alpha +1+k}\right)\,A_L\,,\) 
where
\(\qquad \quad A_L=\left(\frac{\sqrt{\varepsilon }}{1-\varepsilon }\right)^k\left(\frac{1-\varepsilon }{1+\varepsilon }\right)^{\frac{\gamma }{2\sqrt{\varepsilon }}}\sum _{s=0}^k \left(\begin{array}{c}k\\s\end{array}\right)\frac{\Gamma (\frac{\gamma }{2 \sqrt{\varepsilon }} + s)\,\Gamma (\frac{\gamma }{2 \sqrt{\varepsilon }}+1)}{\Gamma (\frac{\gamma }{2 \sqrt{\varepsilon }})\,\Gamma (\frac{\gamma }{2 \sqrt{\varepsilon }}-k+s+1)}\left(\frac{1-\varepsilon }{1+\varepsilon }\right)^s\,.\) 

It turns out the numbers \(q_R\)  and \(q_L\)  given by Proposition
REF  have a limit when \(\sqrt{\varepsilon }\rightarrow 0\) .
Lemma 3.5  
Assume that the condition (REF ) holds. Then for each fixed \(\alpha \)  the numbers \(q_R\)  and \(q_L\)  computed by
Proposition
REF  satisfy the limits
\(\,\lim _{\sqrt{\varepsilon } \rightarrow 0} q_R=-(-1)^{\frac{\alpha }{2}}\,\infty ,\qquad \lim _{\sqrt{\varepsilon } \rightarrow 0} q_L=(-1)^{\frac{\alpha }{2}}\,\infty \,\) 
when \(\sqrt{\varepsilon } \rightarrow 0\) .

Applying the limit
\(\lim _{|z| \rightarrow \infty }\frac{\Gamma (z+a)}{\Gamma (z)\,z^a}=1\) 
we find that
\(q_R &\rightarrow &-\left(\frac{1+\varepsilon }{1-\varepsilon }\right)^{\frac{\gamma }{2 \sqrt{\varepsilon }}}\frac{(-1)^{\frac{\alpha }{2}}}{(2 \sqrt{\varepsilon })^{\alpha -1}\,\Gamma (\frac{\alpha }{2})}\sum _{k=0}^{\frac{\alpha -2}{2}}\frac{(-1)^k\,\Gamma (\alpha -1-k)}{k!\,(\frac{\alpha -2}{2}-k)!}\left(\frac{2 \sqrt{\varepsilon }\,\gamma }{1-\varepsilon ^2}\right)^k,\\[0.2ex]q_L &\rightarrow &\left(\frac{1-\varepsilon }{1+\varepsilon }\right)^{\frac{\gamma }{2 \sqrt{\varepsilon }}}\frac{(-1)^{\frac{\alpha }{2}}}{(2 \sqrt{\varepsilon })^{\alpha -1}\,\Gamma (\frac{\alpha }{2})}\sum _{k=0}^{\frac{\alpha -2}{2}}\frac{\Gamma (\alpha -1-k)}{k!\,(\frac{\alpha -2}{2}-k)!}\left(\frac{2 \sqrt{\varepsilon }\,\gamma }{1-\varepsilon ^2}\right)^k\) 
when \(\sqrt{\varepsilon } \rightarrow 0\) .
Now the statement follows from the observation that \(q_R\)  and \(q_L\)  are expressed as finite sums and from the
limits
\(\lim _{\sqrt{\varepsilon } \rightarrow 0}\left(\frac{1+\varepsilon }{1-\varepsilon }\right)^{\frac{\gamma }{2 \sqrt{\varepsilon }}}=1,\quad \lim _{\sqrt{\varepsilon } \rightarrow 0}\left(\frac{1-\varepsilon }{1+\varepsilon }\right)^{\frac{\gamma }{2 \sqrt{\varepsilon }}}=1\,.\) 
\(\Box \)
From Lemma REF  it follows that the sign of the limit of the number \(q_k, k=L, R\)  depends on the parameter
\(\alpha \)  but we always have that
\(\,\lim _{\sqrt{\varepsilon } \rightarrow 0} (q_R + q_L)=\infty - \infty \,.\,\) 
Remark 3.6 
The result of Lemma REF  is the identification mark of the unfolding of a resonant regular point.
Recall that in our previous works [12]}, [13]} all the coefficients place before the logarithmic terms in the solution
of the unfolded equation have a finite limit when \(\sqrt{\varepsilon } \rightarrow 0\) .
But in all previous cases these logarithmic terms measure how the Stokes matrices of the initial equation are transformed
to the monodromy matrices of the unfolded equation. This time the logarithmic terms in the solution of the unfolded
equation correspond to an unfolding of the monodromy matrix of the initial equation to two monodromy matrices of the
unfolded equation.

 The unfolded monodromy around the origin 
In this section we connect by a radial limit \(\sqrt{\varepsilon } \rightarrow 0\)  the monodromy
matrices \(M_j(\varepsilon ), j=R, L\)  of the unfolded equation with the monodromy matrix \(M_0\) 
of the DCHE.
Thanks to Proposition
REF  we have the following property of the fundamental matrices
\(\Phi _0(x, 0)\)  and \(\Phi (x, \varepsilon )\) .
Corollary 3.7  
The fundamental matrix \(\Phi _0(x, \varepsilon )\)  of the unfolded equation given by Theorem REF  converges uniformly on compact
sets of \(D_R\)  to the fundamental matrix \(\Phi _0(x, 0)\)  of the DCHE given by Theorem REF 
when \(\sqrt{\varepsilon } \rightarrow 0\) .

<FIGURE>Let \(\tilde{D}_R\)  be the universal covering of the disk \(D_R\) . Consider the DCHE
and its local fundamental matrix solution
\(\Phi _0(x, 0)\)  on \(\tilde{D}_R\) . Let \(\gamma \in D_R\)  be a closed loop starting and ending at the same point
\(x_0\in D_R \backslash {\mathbb {R}}\)  as in section 3.2,
encircling the origin and the points \(x_R\)  and \(x_L\)  and oriented counter-clockwise as in Figure REF .
Let \(\tilde{\gamma }\)  be the lift of the loop \(\gamma \)  on
\(\tilde{D}_R\) . The fundamental matrix solution \(\Phi _0(x, 0)\)  of the DCHE introduced in Theorem REF 
is a holomorphic multi-valued function on \(\tilde{D}_R\) . The analytic continuation of \(\Phi _0(x, 0)\)  along
\(\tilde{\gamma }\)  leads to a new fundamental matrix solution \([\Phi _0(x, 0)]_{\tilde{\gamma }}\)  of the DCHE.
The connection between these two fundamental matrix solutions is given by the monodromy matrix
\(M_0=e^{2 \pi \,i\,\Lambda }\,e^{2 \pi \,i\,J}=e^{2 \pi \,i\,J}\)  from
(REF )
\(\,\left[\Phi _0(x, 0)\right]_{\tilde{\gamma }}=\Phi _0(x, 0)\,M_0\,.\,\) 
Let \(D_R(\varepsilon )\)  be a disk obtained from \(D_R\)  by making a cut between the points \(x_L\)  and \(x_R\)  through the real
axis. The origin belongs to this cut. When \(\sqrt{\varepsilon } \rightarrow 0\)  the disk \(D_R(\varepsilon )\)  tends to the disk \(D_R\) .
Let \(\tilde{D}_R(\varepsilon )\)  be the universal covering of \(D_R(\varepsilon )\) . Let
\(\tilde{\gamma }(\varepsilon )\in \tilde{D}_R(\varepsilon )\)  be the unfolding of the loop \(\tilde{\gamma }\)  in the following sense.
The loop \(\tilde{\gamma }(\varepsilon )\)  starts with the same point \(x_0\) . Then it goes consecutively round \(x_L\)  firstly
in the positive sense and then it goes round \(x_R\)  again in the positive sense (see). In this way the projection \(\gamma (\varepsilon )\)  of
\(\tilde{\gamma }(\varepsilon )\)  on \(D_R(\varepsilon )\)  is a closed loop that belongs to the homotopy class \([\gamma ]\)  of the
loop \(\gamma \)  (see Figure REF ). Analytic continuation of the fundamental matrix \(\Phi _0(x, \varepsilon )\)  along \(\tilde{\gamma }(\varepsilon )\) 
yields a new fundamental matrix \(\left[\Phi _0(x, \varepsilon )\right]_{\tilde{\gamma }(\varepsilon )}\) .
The connection between these two fundamental matrices is measured geometrically by an invertible constant
matrix \(M_0(\varepsilon )\)
\(\left[\Phi _0(x, \varepsilon )\right]_{\tilde{\gamma }(\varepsilon )}=\Phi _0(x, \varepsilon )\,M_0(\varepsilon )\,.\) 
Definition 3.8 
We call the invertible matrix \(M_0(\varepsilon )\)  defined by (REF ) the unfolded monodromy matrix
around the origin.

<FIGURE>The reducibility ensures the connection by a limit \(\sqrt{\varepsilon } \rightarrow 0\)  between the fundamental matrix solutions
\(\Phi _0(x, 0)\)  and \(\Phi _0(x, \varepsilon )\) , which ensures such a connection between the monodromy around the origin
and the unfolded monodrmy
Theorem 3.9  
The unfolded monodromy matrix \(M_0(\varepsilon )\)  around the origin depends analytically on \(\sqrt{\varepsilon }\) 
and converges when \(\sqrt{\varepsilon } \rightarrow 0\)  to the monodromy matrix around the origin \(M_0\) 
defined by (REF ).

Since the fundamental matrix \(\Phi _0(x, \varepsilon )\)  converges uniformly on the compact sets
of \(D_R\) 
to the fundamental matrix \(\Phi _0(x, 0)\) , so does the fundamental matrix
\(\left[\Phi _0(x, \varepsilon )\right]_{\tilde{\gamma }(\varepsilon )}\)  to the fundamental matrix
\(\left[\Phi (x, 0)\right]_{\tilde{\gamma }}\) .
Then the matrix \(M_0(\varepsilon )\)  must converge to the monodromy matrix \(M_0\)  when
\(\sqrt{\varepsilon } \rightarrow 0\) .
\(\Box \)
When the origin is a resonant singularity we find
that the unfolded monodromy matrix \(M_0(\varepsilon )\)  is expressed in terms of the monodromy
matrices \(M_L(\varepsilon )\)  and \(M_R(\varepsilon )\) .
Theorem 3.10  
Assume that the condition (REF ) holds.
Let \(M_j(\varepsilon ), j=L, R\)  and \(M_0(\varepsilon )\)  be the monodromy matrices and the unfolded
monodromy matrix of the unfolded equation with respect to the fundamental matrix solution \(\Phi _0(x, \varepsilon )\) 
Then they
satisfy the following relation
\(M_0(\varepsilon )=M_L(\varepsilon )\,M_R(\varepsilon )=M_R(\varepsilon )\,M_L(\varepsilon )\,.\) 

The proof follows from the fact that \(\gamma _L\,\gamma _R=\gamma \) , where \(\gamma _L\)  and \(\gamma _R\)  are the loops
from section 3.2.
\(\Box \)
As an immediate consequence we have
Corollary 3.11  
Assume that the condition (REF ) holds.
Then the unfolded monodromy matrix \(M_0(\varepsilon )\)  and the matrices
\(e^{2 \pi \,i\,J_j(\varepsilon )}, j=L, R\)  satisfy the following relation
\(\,M_0(\varepsilon )=e^{2 \pi \,i\,J_L(\varepsilon )}\,e^{2 \pi \,i\,J_R(\varepsilon )}=e^{2 \pi \,i\,J_R(\varepsilon )}\,e^{2 \pi \,i\,J_L(\varepsilon )}\,.\,\) 

The statement follows immediately from (REF ) and (REF ).
\(\Box \)
Combining Corollary REF  and Theorem REF  we have that
Proposition 3.12  
Assume that the condition (REF ) holds. Then the matrices \(J_k(\varepsilon ), k=L, R\) 
of the unfolded equation and the monodromy matrix \(M_0\)  of the DCHE are connected
by the limit
\(\,e^{2 \pi \,i\,J_L(\varepsilon )}\,e^{2 \pi \,i\,J_R(\varepsilon )}=e^{2 \pi \,i\,J_R(\varepsilon )}\,e^{2 \pi \,i\,J_L(\varepsilon )}\longrightarrow M_0\,\) 
when \(\sqrt{\varepsilon } \rightarrow 0\) .

Thanks to Proposition
REF  we find the limit of \(q_R+q_L\)  when \(\sqrt{\varepsilon } \rightarrow 0\) .
Corollary 3.13  
Assume that the condition (REF ) holds. Then
\(\lim _{\sqrt{\varepsilon } \rightarrow 0} q_R + q_L=\lambda \,,\) 
where \(\lambda \)  is given by (REF ).

In the Appendix we demonstrate by a direct computation that the limit (REF ) is valid for lower values of the
parameter \(\alpha \) .
Unfolding of the Stokes matrix at the origin 
Throughout this section we assume that \(\beta \ne 0\)  and therefore the origin is an irregular singularity for the DCHE.
In [13]} we have shown by a direct computation, that when \(\alpha =2\) 
the Stokes matrix at the origin of the DCHE (REF ) can be obtained by
a limit of this part of the monodromy matrices around resonant singular points
that governs the existence of the
logarithmic term in the solution of the unfolded equation.
In the section 4.1 we show that this result remains valid for every reducible DCHE (REF ).
In fact the realization of the Stokes matrix as a limit of the part of the monodromy matrix of the unfolded equation
is an effect of the recent theoretical result of Hurtubise, Klimeš, Lambert and Rousseau [5]}, [10]}, [6]}, [7]}.
Using the obtained connection between the analytic invariants of both equations we provide the
Stokes matrix at the origin without studying in details the DCHE. Instead we deal with the perturbed
equation and its monodromy matrices around a resonant singularity.
In the section 4.2 we build explicitly an actual fundamental matrix solution of
the DCHE (REF ) at the origin with respect to which the Stokes matrix has the form obtained in the firs part.
As in the previous section we denote by \(\Phi _0(x, 0)\)  and \(\Phi _0(x, \varepsilon )\)  the fundamental matrices from
(REF ) with the chosen in Section 2 paths \(\Gamma (x, 0)\)  and \(\Gamma (x, \varepsilon )\) .
The Stokes matrix at the origin as a limit of the monodromy matrix around \(x_L\)
Following Lambert and Rousseau [6]}, [7]} we cover the disc \(D_R\)  by two open sectors
\(\Omega _1=\Omega _1(\rho , \kappa ) &=&\left\lbrace  x=r\,e^{i \delta }\,|\,0 < r < \rho ,\, -\kappa -\pi < \delta < \kappa \right\rbrace ,\\[0.1ex]\Omega _2=\Omega _2(\rho , \kappa ) &=&\left\lbrace  x=r\,e^{i \delta }\,|\,0 < r < \rho ,\, -\kappa < \delta < \kappa + \pi \right\rbrace \,,\) 
where \(0 < \kappa < \frac{\pi }{2}\) . The radius \(\rho > R\)  is so chosen that
\(x_{LL}, x_{RR} \notin \Omega _1 \cup \Omega _2\) . Denote by \(\Omega _R\)  and \(\Omega _L\)  the connected components of the
intersection \(\Omega _1\cap \Omega _2\) , as \(x_R\in \Omega _R,\,x_L\in \Omega _L\) . Let \(\widetilde{D}_R\)  be the universal
covering of the disk \(D_R\) . Let \(\widetilde{\Omega }_1, \widetilde{\Omega }_2\)  and \(\widetilde{\Omega }_j, j=R, L\)  be the lifts
of the sectors \(\Omega _1, \Omega _2\)  and \(\Omega _j, j=R, L\)  on \(\widetilde{D}_R\) .
<FIGURE>Consider the DCHE (REF ) over \(\widetilde{\Omega }_1 \cup \widetilde{\Omega }_2\) .
From the sectorial normalization theorem of Sibuya [37]} and the theorem of Hukuhara-Turrittin[38]} it follows that
the actual fundamental matrix solution \(\Phi _0(x, 0)\)  of the DCHE can be represented as
\(\,\Phi _j(x, 0)=\exp (G x)\,H_j(x)\,\left[x^{\Lambda }\,\exp \left(-\frac{B}{x}\right)\right]_j\,\) 
on the sectors \(\widetilde{\Omega }_j,\,j=1, 2\) , recpectively. Here
\(G= {\mathrm {diag}} (0, \gamma ),\quad \Lambda = {\mathrm {diag}} (0, -\alpha ), \quad B= {\mathrm {diag}} (0, \beta )\) 
and \([x^{\Lambda }\,\exp (-B/x)]_j\)  is the branch of the matrix \(x^{\Lambda }\,\exp (-B/x)\)  on \(\widetilde{\Omega }_j, j=L, R\) ,
respectively.
The matrices \(H_j(x)\)  are holomoprhic matrix functions on \(\widetilde{\Omega }_j\) , respectively, as both of them are asymptotic
in the Gevrey 1 sense to the same formal matrix \(\hat{H}(x)\)  on \(\widetilde{\Omega }_j, j=L, R\) . On the sector \(\widetilde{\Omega }_R\) 
the fundamental matrix solutions \(\Phi _1(x, 0)\)  and \(\Phi _2(x, 0)\)  coincide.
On the sector \(\widetilde{\Omega }_L\)  the jump of the solution \(\Phi _2(x, 0)\)  to the solution \(\Phi _1(x, 0)\)  is measured geometrically
by the Stokes matrix \(St_{\pi }\) , corresponding to the singular direction \(\theta =\pi \)
\(\Phi _2(x, 0)=\Phi _1(x, 0)\,St_{\pi }\,,\) 
where
\(\,St_{\pi }=\left(\begin{array}{cc}1 & \mu \\0 & 1\end{array}\right)\,.\,\) 
Consider now the unfolded equation. The next theorem describes the behavior of the fundamental matrix solution \(\Phi _0(x, \varepsilon )\) 
near the singular points when \(x_L\)  is a resonant singularity.
Theorem 4.1  
Assume that the condition (REF ) holds. Then
the fundamental matrix solution \(\Phi _0(x, \varepsilon )\)  of the unfolded equation depends analytically on \(\sqrt{\varepsilon }\)  and it is represented
in a neighborhood of the resonant singularity \(x_L\)  which does not contain the point \(x_R\)  as
\(\Phi _0(x, \varepsilon )=\left(I_L(\varepsilon ) + \mathcal {O}(x-x_L)\right)\,(x-x_L)^{\frac{1}{2} \Lambda + \frac{1}{2 x_L}\,B}\,(x-x_L)^{T_L(\varepsilon )}\) 
and in a neighborhood of the non-resonant singularity \(x_R\)  which does not contain the point \(x_L\)  as
\(\,\Phi _0(x, \varepsilon )=\left(I_R(\varepsilon ) + \mathcal {O}(x-x_R)\right)\,(x-x_R)^{\frac{1}{2} \Lambda + \frac{1}{2 x_R}\,B}\,.\,\) 
The matrices \(I_j(\varepsilon ) + \mathcal {O}(x-x_j),\,j=L, R\)  are holomorphic matrix functions there.
The matrices \(\Lambda \)  and \(B\)  are given by (REF ). The matrix \(T_L(\varepsilon )\)  is defined as
\(\,T_L(\varepsilon )=\left(\begin{array}{cc}0 & d_L\\0 & 0\end{array}\right)\,,\,\) 
where
\(\,d_L= {\mathrm {Res}} (w_2^{\prime }(x, \varepsilon ),\,x=x_L)\,.\,\) 

The proof is similar to the proof of Proposition 4.7 in [12]}.
\(\Box \)
With respect to the matrix solution \(\Phi _0(x, \varepsilon )\)  from Theorem REF  the local monodromy matrices
\(M_j(\varepsilon )\) 
around the singular point \(x_j, j=L, R\)  have the form
\(\,M_L(\varepsilon )=e^{2\pi \,i\,(\frac{1}{2}\,\Lambda + \frac{1}{2\,x_L}\,B)}\,e^{2 \pi \,i\,T_L(\varepsilon )}=e^{2 \,\pi \,T_L(\varepsilon )},\qquad M_R(\varepsilon )=e^{2\pi \,i\,(\frac{1}{2}\,\Lambda + \frac{1}{2\,x_R}\,B)}\,.\,\) 
Let \(\Omega _1(\varepsilon )\)  and \(\Omega _2(\varepsilon )\)  be the sectors obtained from the sectors \(\Omega _1\)  and \(\Omega _2\) 
by making a cut between the points \(x_L\)  and \(x_R\)  through the real axis (see Figure REF ).
The origin belongs to this cut. When
\(\sqrt{\varepsilon } \rightarrow 0\)  the sectors \(\Omega _j(\varepsilon )\)  tend to the sectors \(\Omega _j, j=L, R\) , respectively.
Let \(\widetilde{\Omega }_1(\varepsilon )\)  and \(\widetilde{\Omega }_2(\varepsilon )\)  be the lifts of the sectors
\(\Omega _1(\varepsilon )\)  and \(\Omega _2(\varepsilon )\)  on \(\widetilde{D}_R\) .
Consider the unfolded equation over \(\widetilde{\Omega }_1(\varepsilon ) \cup \widetilde{\Omega }_2(\varepsilon )\) .
The fundamental matrix solution \(\Phi _0(x, \varepsilon )\)  writes also as
\(\Phi _0(x, \varepsilon )=G(x, \varepsilon )\,H(x, \varepsilon )\,(x-x_L)^{\frac{1}{2}\,\Lambda + \frac{1}{2\, x_L}\,B}\,(x-x_R)^{\frac{1}{2}\,\Lambda + \frac{1}{2\, x_R}\,B}\,,\) 
where
\(\,G(x, \varepsilon )=(x-x_{LL})^{-\frac{x_{LL}}{2}\,G}\,(x_{RR} -x)^{-\frac{x_{RR}}{2}\,G}\,\,\) 
and
\(\,H(x, \varepsilon )=\left(\begin{array}{cc}1 & (x-x_L)^{-\frac{\beta }{2 \sqrt{\varepsilon }}+ \frac{\alpha }{2}}\,(x-x_R)^{\frac{\beta }{2 \sqrt{\varepsilon }}+ \frac{\alpha }{2}}\,w_2(x, \varepsilon )\\[0.1ex]0 & 1\end{array}\right)\,.\,\) 
The next proposition that follows immediately from Proposition
REF  is a key for the extension of the results in [13]}
Proposition 4.2  
The fundamental matrix solution \(\Phi _0(x, \varepsilon )\)  from (REF ) converges uniformly on compact sets of
\(\Omega _R \cup \Omega _L\)  to the actual fundamental matrix solution \(\Phi _0(x, 0)\)  at the origin of the DCHE when
\(\sqrt{\varepsilon } \rightarrow 0\) .

<FIGURE>As in [13]} Proposition
REF  allows us to identify the so called unfolded Stokes matrix \(St_L(\varepsilon )\)  with the
matrix \(e^{2 \pi \,i\,T_L}\)  when the point \(x_L\)  is a resonant singularity
Corollary 4.3  \((\) Corollary 6.2 in [13]}\()\) 
Assume that the condition (REF ) holds. Then
\(\,St_L(\varepsilon )=e^{2\,\pi \,i\,T_L(\varepsilon )}\,.\,\) 

Let us compute the limit of the matrix \(T_L(\varepsilon )\)  when \(\sqrt{\varepsilon } \rightarrow 0\) .
The next proposition and lemma gives us the number \(d_L\) and its limit when \(\sqrt{\varepsilon } \rightarrow 0\) .
Proposition 4.4  
Assume that the condition (REF ) holds. Then
\(d_L &=&\frac{1}{\left(\frac{\beta }{2 \sqrt{\varepsilon }}+\frac{\alpha }{2}-1\right)!}\sum _{k=0}^{\frac{\beta }{2 \sqrt{\varepsilon }} + \frac{\alpha }{2}-1}\left(\begin{array}{c}\frac{\beta }{2 \sqrt{\varepsilon }} + \frac{\alpha }{2} -1\\[0.1ex]k\end{array}\right)\,\frac{\Gamma \left(\frac{\beta }{2 \sqrt{\varepsilon }}-\frac{\alpha }{2}+1\right)(-2 \sqrt{\varepsilon })^{-\alpha +1+k}}{\Gamma (2+k-\alpha )}\,A_L\,,\) 
where \(A_L\)  is given by (REF ).

Lemma 4.5  
Assume that the condition (REF ) holds. Then
\(\,\lim _{\sqrt{\varepsilon } \rightarrow 0} d_L=(-\beta )^{1-\alpha }\,\sum _{k=0}^{\infty }(-1)^k\,\frac{\beta ^k\,\gamma ^k}{k!\,\Gamma (2+k-\alpha )}\,.\,\) 

Applying the limit (REF ) for \(z=\frac{\gamma }{2 \sqrt{\varepsilon }}\)  we find that \(A \rightarrow \gamma ^k\) 
when \(\sqrt{\varepsilon } \rightarrow 0\) . Again applying the limit (REF ) for \(z=\frac{\beta }{2 \sqrt{\varepsilon }}\)  and using the limit
(REF ) we obtain the limit of \(d_L\) .
\(\Box \)
In [7]} Lambert and Rousseau prove that the unfolded Stokes matrix \(St_L(\varepsilon )\)  depends
analytically on \(\sqrt{\varepsilon }\)  and tends to the Stokes matrix \(St_{\pi }\)  when \(\sqrt{\varepsilon } \rightarrow 0\) .
Now we will use this fact to obtain by a limit procedure the Stokes matrix \(St_{\pi }\)  from the matrix
\(e^{2 \pi \,i\,T_L(\varepsilon )}\) .
Theorem 4.6  
Assume that \(\beta > 0 \) . Then there exists an actual fundamental matrix solution at the origin of the DCHE
with respect to which the corresponding Stokes matrix \(St_{\pi }\)  is given by
\(\,St_{\pi }=\left(\begin{array}{cc}1 & \mu \\0 & 1\end{array}\right)\,,\,\) 
where
\(\mu = 2\,\pi \,i\, (-\beta )^{1-\alpha }\,\sum _{k=0}^{\infty }(-1)^k\,\frac{\beta ^k\,\gamma ^k}{k!\,\Gamma (2+k-\alpha )}\,.\) 

Actual fundamental matrix solution at the origin of the DCHE 
In this paragraph we present explicitly the actual fundamental matrix solution \(\Phi _0(x, 0)\)  at the origin of the DCHE
with respect to which the Stokes matrix \(St_{\pi }\)  has the form fixed by Theorem REF .
When \(\alpha \in {\mathbb {Z}}\)  we apply the Borel-Laplace summation in order to build this actual solution (see [44]}, [45]} for details).
When \(\alpha \notin {\mathbb {Z}}\)  we directly express the solution \(w_2(x, 0)\)  in terms of Laplace integrals without using the summability
theory. The second approach is more general and also can be used when \(\alpha \in {\mathbb {Z}}\) . But the first approach allows us to
distinguish special solutions of DCHE that are holomorphic in whole \({\mathbb {C}}^*\)  even when \(\alpha \notin {\mathbb {Z}}\) .
We start by building a formal fundamental matrix solution at the origin.
Theorem 4.7  
Assume that \(\beta > 0\) . Then the DCHE (REF ) possesses an unique formal fundamental matrix solution \(\hat{\Phi }_0(x, 0)\) 
at the origin in the form
\(\,\hat{\Phi }_0(x, 0)=\exp (G x)\,\hat{H}(x)\,x^{\Lambda }\,\exp \left(-\frac{B}{x}\right)\,,\,\) 
where the matrices \(G, \Lambda \)  and \(B\)  are given by (REF ).
The matrix \(\hat{H}(x)\)  is defined as follows:

 
If \(\alpha \notin {\mathbb {Z}}\)  then
\(\,\hat{H}(x)=\left(\begin{array}{cc}1 & \frac{x\,\hat{\varphi }(x)}{\beta }\\[0.1ex]0 & 1\end{array}\right)\,\) 
where
\(\hat{\varphi }(x)=\sum _{n=0}^{\infty }\frac{(-1)^n\,S_n\,\Gamma (2-\alpha +n)}{\beta ^n}\,x^{n+1}\,.\) 
Here \(S_n\)  is the \(n\) -th partial sum of the absolutely convergent number series
\(\,S=\sum _{k=0}^{\infty }\frac{(-1)^k\,\beta ^k\,\gamma ^k}{k!\,\Gamma (2-\alpha +k)}\,.\,\) 
In particular,

 
If \(S=0\)  then the power series \(\hat{\varphi }(x)\)  is convergent.

 
If \(S \ne 0\)  then the power series \(\hat{\varphi }(x)\)  is divergent.


 
If \(\alpha \in {\mathbb {N}}\)  then
\(\,\hat{H}(x)=\left(\begin{array}{cc}1 & \frac{x^{\alpha }\,\gamma ^{\alpha -1}\,\hat{\psi }(x)}{\beta } + x^{\alpha }\,P\left(\frac{1}{x}\right)\\[0.1ex]0 & 1\end{array}\right)\,\) 
where
\(\hat{\psi }(x)=\sum _{n=0}^{\infty }\frac{(-1)^n\,W_n\,n!}{\beta ^n}\,x^{n+1}\) 
and \(P\left(\frac{1}{x}\right)\)  is a polynomial in \(\frac{1}{x}\)  of degree \(\alpha -2\)  for \(\alpha \ge 2\)  and
\(P\equiv 0\)  for \(\alpha =1\) .
Here \(W_n\)  is the \(n\) -th partial sum of the absolutely convergent number series
\(\,W=\sum _{k=0}^{\infty }\frac{(-1)^k\,\beta ^k\,\gamma ^k}{k!\,\Gamma (\alpha +k)}\,.\,\) 
In particular,

 
If \(W=0\)  then the power series \(\hat{\psi }(x)\)  is convergent.

 
If \(W\ne 0\)  then the power series \(\hat{\psi }(x)\)  is divergent.


 
If \(\alpha \in {\mathbb {Z}}_{\le 0}\)  then
\(\,\hat{H}(x)=\left(\begin{array}{cc}1 & \frac{x\,\hat{\phi }(x)}{\beta }\\[0.1ex]0 & 1\end{array}\right)\,\) 
where
\(\hat{\phi }(x)=\sum _{n=0}^{\infty }\frac{(-1)^n\,Q_n\,\Gamma (2-\alpha +n)}{\beta ^{1-\alpha +n}}\,x^{n+1}\,.\) 
Here \(Q_n\)  is the \(n\) -th partial sum of the absolutely convergent number series
\(\,Q=\sum _{k=0}^{\infty }\frac{(-1)^k\,\gamma ^k\,\beta ^{1-\alpha +k}}{k!\,\Gamma (2-\alpha +k)}\,.\,\) 
In particular,

 
If \(Q=0\)  then the power series \(\hat{\phi }(x)\)  is convergent.

 
If \(Q\ne 0\)  then the power series \(\hat{\phi }(x)\)  is divergent.



The proof is similar to the proof of Proposition 4.2 in [13]}. We only note that when \(\alpha \notin {\mathbb {Z}}\)  we reduce the solution
\(w_2(x, 0)\)  from (REF ) to the integral \(\int _0^x \frac{e^{-\frac{\beta }{z}}}{z^{\alpha }}\, d z\)  while when \(\alpha \in {\mathbb {Z}}\) 
to the integral \(\int _0^x \frac{e^{-\frac{\beta }{z}}}{z}\,d z\) .
\(\Box \)
The application of summability theory
to the differential equations ensures that the divergent power series \(\hat{\varphi }(x), \hat{\psi }(x)\)  and \(\hat{\phi }(x)\)  are 1-summable
in any direction \(\theta \)  except for the singular direction \(\theta =\pi \) .
Lemma 4.8  
1. Assume that \(W \ne 0\) . Then for any direction \(\theta \ne \pi \)  the function
\(\,\psi _{\theta }(x)=\beta \,\int _0^{+\infty \,e^{i \theta }}\frac{v(\xi )\,e^{-\frac{\xi }{x}}}{\xi + \beta }\,d \xi \,\) 
defines the 1-sum of the power series \(\hat{\psi }(x)\)  from (REF ) in such a direction. Here
\(\,v(\xi )=\sum _{n=0}^{\infty } \frac{\gamma ^n\,\xi ^n}{n!\,\Gamma (\alpha +n)}\,.\,\) 
2.  Assume that \(Q \ne 0\) . Then for every direction \(\theta \ne \pi \)  the function
\(\,\phi _{\theta }(x)=\frac{\beta }{x^{1-\alpha }}\int _0^{+\infty e^{i \theta }}\frac{\xi ^{1-\alpha }\,q(\xi )\,e^{-\frac{\xi }{x}}}{\xi +\beta }\,d \xi \,\) 
defines the 1-sum of the power series \(\hat{\phi }(x)\)  from (REF ) is such a direction.
Here
\(\,q(\xi )=\sum _{k=0}^{\infty } \frac{\gamma ^k\,\xi ^k}{k!\,\Gamma (2-\alpha +k)}\,.\,\) 
The functions \(\psi _{\theta }(x)\)  and \(\phi _{\theta }(x)\)  are holomorphic functions in the open disc
\(\mathcal {D}_{\theta }(|\gamma |)=\left\lbrace x\in {\mathbb {C}}^*\,|\, Re \,\left(\frac{e^{i \theta }}{x}\right) > |\gamma |\right\rbrace \,.\) 

We will prove the second statement of the Lemma. The first is proved in a similar way.
Let \(\alpha \in {\mathbb {Z}}_{\le 0}\) . The formal Borel transform of order 1 of the series \(\hat{\phi }(x)\)  from
(REF ) yields the convergent power series near the origin \(\xi =0\)  of the Borel \(\xi \) -plane
\(\,\phi (\xi )=(\hat{{\mathcal {B}}}_1 \hat{\phi })(\xi )=\sum _{n=0}^{\infty } (-1)^n\,Q_n\,\frac{(1-\alpha +n)!}{\beta ^{1-\alpha +n}}\,\frac{\xi ^n}{n!}\,.\,\) 
The series \(\phi (\xi )\)  can be regarded as the \(1-\alpha \) -th derivative of the series
\(\,w(\xi )=\sum _{n=0}^{\infty } (-1)^n\,Q_n\,\frac{\xi ^{1-\alpha +n}}{\beta ^{1-\alpha +n}}\,.\,\) 
Then from Lemma 6.12 in [13]} it follows that the Laplace transform of order 1 of the function
\(\phi (\xi )\)  is expressed by the Laplace transform of the function \(w(\xi )\)
\(\,({\mathcal {L}}_1 \phi )(x)=\frac{1}{x^{1-\alpha }}\,({\mathcal {L}}_1 w)(x)\,\) 
since \(\frac{d^k\,w}{d\, \xi ^k}=0\)  for \(0 \le k \le -\alpha \) . It turns out that the power series
\(w(\xi )\)  is the Maclaurin series of a well known function. More precisely,
\(\,w(\xi )=(-\beta )^{1-\alpha }\,\left(\frac{1}{1+\frac{\xi }{\beta }} - \sum _{k=0}^{-\alpha } (-1)^k\,\frac{\xi ^k}{\beta ^k}\right)\,q(\xi )\,,\,\) 
where \(q(\xi )\)  is the analytic function in \({\mathbb {C}}\)  defined by the power series
\(\,q(\xi )=\sum _{k=0}^{\infty } \frac{\gamma ^k\,\xi ^k}{k!\,\Gamma (2-\alpha +k)}\,.\,\) 
Since
\(\,\sum _{k=0}^{-\alpha } (-1)^k\frac{\xi ^k}{\beta ^k}=\frac{1-(-1)^{1-\alpha }\frac{\xi ^{1-\alpha }}{\beta ^{1-\alpha }}}{1+\frac{\xi }{\beta }}\,\) 
then
\(({\mathcal {L}}_1 \phi )(x) &=&\frac{(-\beta )^{1-\alpha }}{x^{1-\alpha }}\,\left({\mathcal {L}}_1 \frac{q(\xi )}{1+\frac{\xi }{\beta }}\right)(x) -\frac{(-\beta )^{1-\alpha }}{x^{1-\alpha }}\,\left({\mathcal {L}}_1 \frac{q(\xi )}{1+\frac{\xi }{\beta }}\right)(x) \\[0.3ex]&+&\frac{1}{x^{1-\alpha }} \left({\mathcal {L}}_1 \frac{\xi ^{1-\alpha }\,q(\xi )}{1+\frac{\xi }{\beta }}\right)(x)=\frac{\beta }{x^{1-\alpha }}\int _0^{+\infty e^{i \theta }}\frac{\xi ^{1-\alpha }\,q(\xi )\,e^{-\frac{\xi }{x}}}{\xi +\beta }\,d \xi \,.\) 
Thus the function
\(\,\phi _{\theta }(x)=\frac{\beta }{x^{1-\alpha }}\int _0^{+\infty e^{i \theta }}\frac{\xi ^{1-\alpha }\,q(\xi )\,e^{-\frac{\xi }{x}}}{\xi +\beta }\,d \xi \,\) 
gives the 1-sum of the power series \(\hat{\phi }(x)\)  in any direction \(\theta \ne \pi \) .
Since
\(\,\left|\frac{q(\xi )}{\xi +\beta }\right| \le A \sum _{k=0}^{\infty } \frac{|\gamma |^k\,|\xi |^k}{k!}=A\,e^{|\gamma |\,|\xi |}\,\) 
for an appropriate constant \(A > 0\)  the integral \(\phi _{\theta }(x)\)  exists when \(\alpha \in {\mathbb {Z}}_{\le 0}\) 
and defines a holomorphic function in the open disc \(Re\,\left(\frac{e^{i \theta }}{x}\right) > |\gamma |\) .
This ends the proof.
\(\Box \)
Remark 4.9 
Unfortunately till now we can not derive the 1-sum of the power series \(\hat{\varphi }(x)\)  from
(REF ) in an explicit way. The formal Borel transform of this series
\(\,(\hat{{\mathcal {B}}}_1 \hat{\varphi })(\xi )=\sum _{n=0}^{\infty } \frac{(-1)^n\,S_n\,\Gamma (2-\alpha +n)}{\beta ^n}\,\frac{\xi ^n}{n!}\,\) 
is a convergent power series for \(|\xi | < \beta \) . But we can not specify explicitly the function
whose Maclaurin series is \((\hat{{\mathcal {B}}}_1 \hat{\varphi })(\xi )\) . For this reason we use a slightly different
approach to build an actual solution of the DCHE when \(\alpha \notin {\mathbb {Z}}\) .

Remark 4.10 
Let \(I=(-\pi , \pi ) \subset {\mathbb {R}}\) .
When we move the direction \(\theta \in I\)  the holomorphic functions
\(\psi _{\theta }(x)\)  (resp. \(\phi _{\theta }(x)\) ) glue together analytically and define a holomorphic function \(\tilde{\psi }(x)\) 
(resp. \(\tilde{\phi }(x)\) ) on a sector \(\widetilde{\mathcal {D}}\) 
with opening \(> \pi \)
\(\widetilde{\mathcal {D}}=\bigcup _{\theta \in I}\,\widetilde{\mathcal {D}}_{\theta }(|\gamma |)\,,\) 
where \(\widetilde{\mathcal {D}}_{\theta }(|\gamma |)\)  is the lifting of \(\mathcal {D}_{\theta }(|\gamma |)\) 
on the Riemann surface of the natural logarithm. On \(\widetilde{\mathcal {D}}\)  the function \(\tilde{\psi }(x)\)  (resp. \(\tilde{\phi }(x)\) )
is asymptotic to
the power series \(\hat{\psi }(x)\)  (resp. \(\hat{\phi }(x)\) ) in Gevrey 1 sense and defines the 1-sum of this series there. The restriction of
\(\tilde{\psi }(x)\)  (resp. \(\tilde{\phi }(x)\) ) on \({\mathbb {C}}^*\)  is a multivalued function. In every direction \(\theta \ne \pi \)  the function \(\tilde{\psi }(x)\)  (resp. \(\tilde{\phi }(x)\) )
has only one value than coincides with the function \(\psi _{\theta }(x)\)  from Lemma REF (1) (resp. \(\phi _{\theta }(x)\)  from Lemma REF (2)). Near the singular direction
\(\theta =\pi \)  the function \(\tilde{\psi }(x)\)  (resp. \(\tilde{\phi }(x)\) ) has two different values: \(\psi ^{+}_{\pi }(x)=\psi _{\pi +\epsilon }(x)\)  (resp. \(\phi ^{+}_{\pi }(x)=\phi _{\pi +\epsilon }(x)\) ) and
\(\psi ^{-}_{\pi }(x)=\psi _{\pi -\epsilon }(x)\)  (resp. \(\phi ^{-}_{\pi }(x)=\phi _{\pi -\epsilon }(x)\) ) for a small number \(\epsilon > 0\) .

Now we can present an actual fundamental matrix solution at the origin.
Theorem 4.11  
Assume that \(\beta > 0\) .
Then

 
Assume that one of the following conditions holds: \((\alpha \in {\mathbb {N}},\, W=0),\,(\alpha \in {\mathbb {Z}}_{\le 0},\,Q=0)\) 
or \((\alpha \notin {\mathbb {Z}}, S=0)\) . Then the DCHE (REF ) possesses an unique actual fundamental
matrix solution \(\Phi _0(x, 0)\)  at the origin in the form
\(\Phi _0(x, 0)=\exp (G x)\,H(x)\,x^{\Lambda }\,\exp \left(-\frac{B}{x}\right)\,,\) 
where the matrices \(G, \Lambda \)  and \(B\)  are given by (REF ) and \(H(x)\)  is a holomorphic matrix function in whole \({\mathbb {C}}\) .
More precisely, when \(\alpha \notin {\mathbb {Z}}\)  and \(S=0\)  the matrix \(H(x)\)  coincides with the matrix \(\hat{H}(x)\) 
from Theorem REF (1). When
\(\alpha \in {\mathbb {N}}\)  and \(W=0\)  the matrix \(H(x)\)  coincides with the matrix \(\hat{H}(x)\)  from Theorem REF (2).
When \(\alpha \in {\mathbb {Z}}_{\le 0}\)  and \(Q=0\)  the matrix \(H(x)\)  coincides with the matrix \(\hat{H}(x)\)  from Theorem REF (3).

 
Assume that \(Q\ne 0,\,W\ne 0, \,S\ne 0\) .
Then the DCHE (REF ) possesses an unique actual matrix solution \(\widetilde{\Phi }_0(x, 0)\)  at the origin in the form
(REF ) which is a holomorphic matrix function on the sector \(\widetilde{\mathcal {D}}\)  from (REF ) whose opening is
\(>\pi \) . The restriction of \(\widetilde{\Phi }_0(x, 0)\)  on \({\mathbb {C}}^*\)  is a multivalued function.
For any direction \(\theta \ne \pi \)  this solution has only one value
\(\Phi ^{\theta }_0(x, 0)\)  in the form
\(\Phi ^{\theta }_0(x, 0)=\exp (G x)\,H_{\theta }(x)\,x^{\Lambda }\,\exp \left(-\frac{B}{x}\right)\,,\) 
where the matrices \(G, \Lambda \)  and \(B\)  are given by (REF ).
The matrix \(H_{\theta }(x)\)  is defined as follows:

 
If \(\alpha \notin {\mathbb {Z}}\)  then
\(\,H_{\theta }(x)=\left(\begin{array}{cc}1 & \frac{x\,\varphi _{\theta }(x)}{\beta }\\[0.15ex]0 & 1\end{array}\right)\,,\,\) 
where
\(\varphi _{\theta }(x)=\sum _{k=0}^{\infty }\frac{\gamma ^k\,x^k}{k!}\,\left(\int _0^{+\infty e^{i \theta }}\frac{e^{-\frac{\xi }{x}}}{(1+\frac{\xi }{\beta })^{2-\alpha +k}}\,d \xi \right)\,.\) 

 
If \(\alpha \in {\mathbb {N}}\)  then
\(\,H_{\theta }(x)=\left(\begin{array}{cc}1 & \frac{x^{\alpha }\,\gamma ^{\alpha -1}\,\psi _{\theta }(x)}{\beta } + x^{\alpha }\,P\left(\frac{1}{x}\right)\\[0.1ex]0 & 1\end{array}\right)\,\) 
where \(\psi _{\theta }(x)\)  is defined by Lemma REF (1) and extended by Remark REF .

 
If \(\alpha \in {\mathbb {Z}}_{\le 0}\)  then
\(\,H_{\theta }(x)=\left(\begin{array}{cc}1 & \frac{x\,\phi _{\theta }(x)}{\beta }\\[0.1ex]0 & 1\end{array}\right)\,\) 
where \(\phi _{\theta }(x)\)  is defined by Lemma REF (2) and extended by Remark REF .
For the singular direction \(\theta =\pi \)  the DCHE (REF ) possesses two actual fundamental matrix solution at the
origin
\(\,\Phi _0^{\pi \pm }(x, 0)=\Phi _0^{\pi \pm \epsilon }(x, 0)\,,\,\) 
where \(\epsilon > 0\)  is a small number and the matrices \(\Phi _0^{\pi \pm \epsilon }\)  are given by (REF ).



The proof of items \((1), (2.b)\)  and \((2.c)\)  follows directly from the theorem of Hukuhara-Turrittin-Martinet-Ramis , [45]}.
We give the proof of item \((2.a)\) .
Let \(\alpha \notin {\mathbb {Z}}\) . The solution \(w_2(x, 0)\)  from (REF ) becomes
\(\,w_2(x, 0)=\sum _{k=0}^{\infty } \frac{\gamma ^k}{k!}\left(\int _0^x \frac{e^{-\frac{\beta }{z}}}{z^{\alpha -k}}\,d z\right)=e^{-\frac{\beta }{x}}\sum _{k=0}^{\infty } \frac{\gamma ^k}{k!}\left(\int _0^x \frac{e^{-\frac{\beta }{z}+ \frac{\beta }{x}}}{z^{\alpha -k}}\,d z\right)\,.\,\) 
By setting
\(\,- \frac{\beta }{z} + \frac{\beta }{x}=-\frac{\xi }{x}\,\) 
we transform the solution \(w_2(x, 0)\)  into
\(\,w_2(x, 0)=\frac{e^{-\frac{\beta }{x}}\,x^{1-\alpha }}{\beta }\sum _{k=0}^{\infty } \frac{\gamma ^k\,x^k}{k!}\left(\int _0^{+\infty }\frac{e^{-\frac{\xi }{x}}}{(1+\frac{\xi }{\beta })^{2-\alpha +k}}\,d \xi \right)\,.\,\) 
Now we will show that this infinite sum defines a holomorphic function on the open disc
\(\mathcal {D}_{\theta }(|\gamma |)\)  defined by (REF )
for \(\theta \in (-\pi , \pi )\) .
Since
\(1/|1+\frac{\xi }{\beta }| \le 1\)  for \(\cos \, \theta \ge 0\)  and
\(1/|1+\frac{\xi }{\beta }| \le 1/|\sin \,\theta |\)  for \(\cos \, \theta < 0\)  we find that when \(Re\,(\alpha ) \le 2\)
\(\frac{1}{|(1+\frac{\xi }{\beta })^{k-\alpha +2}|} \le \left\lbrace \begin{array}{ccc}A & \textrm {for} & \cos \,\theta \ge 0,\\[1.15ex]\frac{A}{|\sin \,\theta |^{k + 2 - Re\,(\alpha )}} & \textrm {for} & \cos \,\theta < 0\end{array}\right.\) 
where \(A=e^{-(Im \,\alpha )\,\arg (1+\frac{\xi }{\beta })} > 0\) .
Thus in this case each integral can be analytically continued along any ray \(\theta \ne \pi \)  from 0 to \(+\infty \,e^{i \theta }\)  and
defines a holomorphic function in the open disc
\(Re\, \left(\frac{e^{i \theta }}{x}\right) > 0\)  whose opening is \(< \pi \) .
Let \(x\in \mathcal {D}_{\theta }(|\gamma |)\)  and let \(\kappa =\arg (x)\) . Note that from \(x\in \mathcal {D}_{\theta }(|\gamma |)\)  it follows
that \(|x| < \frac{\cos (\theta -\kappa )}{|\gamma |}\) . Then we find that
when \(\cos \,\theta \ge 0\)  and \(Re (\alpha ) \le 2\)
\(\,\left|\sum _{k=0}^{\infty } \frac{\gamma ^k\,x^k}{k!}\left(\int _0^{+\infty \,e^{i \theta }}\frac{e^{-\frac{\xi }{x}}}{(1+\frac{\xi }{\beta })^{2-\alpha +k}}\,d \xi \right)\right|\le \frac{A}{c}\sum _{k=0}^{\infty } \frac{|\gamma |^k\,|x|^k}{k!} <\frac{A}{c}\sum _{k=0}^{\infty } \frac{\cos ^k (\theta -\kappa )}{k!} < \infty \,,\,\) 
where \(c=Re \left(\frac{e^{i \theta }}{x}\right) > |\gamma | > 0\) .
Similarly, when \(cos \,\theta < 0\)  and \(Re (\alpha ) \le 2\)  we have that
\(\left|\sum _{k=0}^{\infty } \frac{\gamma ^k\,x^k}{k!}\left(\int _0^{+\infty \,e^{i \theta }}\frac{e^{-\frac{\xi }{x}}}{(1 + \frac{\xi }{\beta })^{2-\alpha +k}}\,d \xi \right)\right|&\le &\frac{A}{c\,|\sin \theta |^{2- Re (\alpha )}}\sum _{k=0}^{\infty } \frac{|\gamma |^k\,|x|^k}{k!\,|\sin \, \theta |^k} \\[0.2ex]&<&\frac{A}{c\,|\sin \ \theta |^{2-Re (\alpha )}}\sum _{k=0}^{\infty } \frac{\cos ^k (\theta -\kappa )}{k!\,|\sin \theta |^k } < \infty \,.\) 
Thus from the Wiierstrass's theorem it follows that when \(Re (\alpha ) \le 2\)  the functional series
\(\sum _{k=0}^{\infty } \frac{\gamma ^k\,x^k}{k!}\left(\int _0^{+\infty \,e^{i \theta }}\frac{e^{-\frac{\xi }{x}}}{(1+\frac{\xi }{\beta })^{2-\alpha +k}}\,d \xi \right)=\sum _{k=0}^{\infty } \frac{\gamma ^k\,x^k}{k!}\,\varphi ^{\theta }_k(x)\) 
converges uniformly on the compact sets of the open disc \(\mathcal {D}_{\theta }(|\gamma |)\) .
Since for all \(k \ge 0\)  the functions \(\frac{\gamma ^k\,x^k}{k!}\,\varphi ^{\theta }_k(x)\)  are holomorphic functions
on \(\mathcal {D}_{\theta }(|\gamma |)\)  so the sum.
Assume now that there exists \(k_1\in {\mathbb {N}}_0\)  such that
\(k+2 - Re\,(\alpha ) < 0\)  for \(0 \le k \le k_1\)  while \(k_1+1 +2 -Re (\alpha ) \ge 0\) .
Since \(|(1+\frac{\xi }{\beta })^{\alpha -k-2}| \le A\,(1+\frac{|\xi |}{\beta })^{Re (\alpha ) -k-2}\) 
where \(A\)  is above we find that for \(0 \le k \le k_1\)
\(& &\left|\int _0^{+\infty e^{i \theta }}\frac{e^{-\frac{\xi }{x}}}{\left(1+\frac{\xi }{\beta }\right)^{2-\alpha +k}}\,d \xi \right| \le A\,2^{Re (\alpha ) -k-2}\int _0^{\beta e^{i \theta }}e^{-|\xi |\,c}\,d \xi \\[0.2ex]&+&A\,\left(\frac{2}{\beta }\right)^{Re (\alpha ) -k-2}\int _{\beta \,e^{i \theta }}^{+\infty e^{i \theta }}|\xi |^{Re (\alpha ) -k-2}\,e^{-c\,|\xi |}\,d \xi \\[0.2ex]&\le &\frac{2^{Re (\alpha ) -k-2}\,A}{c} (1-e^{-\beta \,c})+\frac{A}{c^{Re (\alpha )-k-1}}\,\left(\frac{2}{\beta }\right)^{Re (\alpha ) -k-2}\,\Gamma (Re (\alpha )-k-1)\,.\) 
Therefore for \(0 \le k \le k_1\)  we also can continue analytically each integral along any ray \(\theta \ne \pi \)  and
define holomorphic functions \(\varphi ^{\theta }_k(x)\)  in the open disc
\(Re\, \left(\frac{e^{i \theta }}{x}\right) > 0\)  whose opening is \(< \pi \) . Then for \(x\in \mathcal {D}_{\theta }(|\gamma |)\) 
we have
\(& &\left|\sum _{k=0}^{\infty } \frac{\gamma ^k\,x^k}{k!}\left(\int _0^{+\infty \,e^{i \theta }}\frac{e^{-\frac{\xi }{x}}}{(1+\frac{\xi }{\beta })^{2-\alpha +k}}\,d \xi \right)\right|\le K \sum _{k=0}^{k_1} \frac{|\gamma |^k\,|x|^k}{2^k\,k!} \\[0.25ex]&+&\frac{A}{c^{Re (\alpha )-1}}\left(\frac{2}{\beta }\right)^{Re (\alpha )-2}\sum _{k=0}^{k_1} \frac{|\gamma |^k\,c^k\,\beta ^k\,\Gamma (Re (\alpha ) - k-1)}{2^k\,k!}\,x^k+ |F_{k_1+1}(x)|\\[0.25ex]&<&K \sum _{k=0}^{k_1} \frac{\cos ^k (\theta -\kappa )}{2^k\,k!} +\frac{A}{c^{Re (\alpha )-1}}\left(\frac{2}{\beta }\right)^{Re (\alpha )-2}\sum _{k=0}^{k_1} \frac{\cos ^k (\theta -\kappa )\,c^k\,\beta ^k\,\Gamma (Re (\alpha ) - k-1)}{2^k\,k!}\\[0.25ex]&+&|F_{k_1+1}(x)|\) 
where \(K=\frac{2^{Re (\alpha ) -2}\,A}{c} (1-e^{-\beta \,c})\) .
For the last addend we have that
\(\,|F_{k_1+1}(x)| \le \frac{A}{c}\sum _{k_1+1}^{\infty } \frac{|\gamma |^k\,|x|^k}{k!} <\frac{A}{c} \sum _{k_1+1}^{\infty } \frac{\cos ^k (\theta -\kappa )}{k!} < \infty \,\) 
when \(\cos \,\theta \ge \)  and
\(\,|F_{k_1+1}(x)| \le \frac{A}{c\,|\sin \, \theta |^{2- Re (\alpha )}}\sum _{k_1+1}^{\infty } \frac{|\gamma |^k\,|x|^k}{k!\,|\sin \,\theta |^k} <\frac{A}{c |\sin \theta |^{2- Re (\alpha )}}\sum _{k_1+1}^{\infty } \frac{\cos ^k (\theta -\kappa )}{k!\,|\sin \,\theta |^k} < \infty \,\) 
when \(\cos \,\theta < 0\) .
As a result the functional series (REF )
defines a holomorphic function on the disc \(\mathcal {D}_{\theta }(|\gamma |)\)  whose opening is \(< \pi \) .
We denote this function by \(\varphi _{\theta }(x)\) .
When we move \(\theta \in (-\pi , \pi )\)  the holomorphic functions \(\varphi _{\theta }(x)\)  glue together
analytically and define a holomorphic function \(\tilde{\varphi }(x)\)  on an open sector
\(\widetilde{\mathcal {D}}\)  from (REF ) whose opening \(> \pi \) .
The restriction of \(\tilde{\varphi }(x)\)  on \({\mathbb {C}}^*\)  is a multivalued function.
For every direction \(\theta \ne \pi \)  it has only one value. Near the singular direction \(\theta =\pi \)  it has two different values:
\(\varphi ^{+}_{\theta }(x)=\sum _{k=0}^{\infty } \frac{\gamma ^k\,x^k}{k!}\,\varphi ^{\theta +\epsilon }_k(x)\)  and
\(\varphi ^{-}_{\theta }(x)=\sum _{k=0}^{\infty } \frac{\gamma ^k\,x^k}{k!}\,\varphi ^{\theta -\epsilon }_k(x)\)  where \(\epsilon > 0\)  is a small number.
This ends the proof.
\(\Box \)
Remark 4.12 
It seems that the function \(\varphi _{\theta }(x)\)  from (REF ) plays the part of 1 sum of the formal series
\(\,\hat{\varphi }(x)=\sum _{k=0}^{\infty }\frac{\gamma ^k\,x^k}{k!}\,\hat{\varphi }_k(x)\,,\,\) 
where
\(\,\hat{\varphi }_k(x)=\sum _{n=0}^{\infty }\frac{(-1)^n\,(2-\alpha +k)_n}{\beta ^n}\,x^{n+1}\,.\,\) 
Here \((a)_n\)  is the Pochhammer symbol.
We are going to discuss in our next work the summation of such a series \(\hat{\varphi }(x)\) .

Theorem 4.13  
Assume that \(\beta > 0\) . Then with respect to the actual fundamental matrix solution \(\Phi _0(x, 0)\) 
defined by Theorem REF  the DCHE (REF ) has a Stokes matrix \(St_{\pi }\)  at the origin in the form
\(\,St_{\pi }=\left(\begin{array}{cc}1 & \mu \\0 & 1\end{array}\right)\,,\,\) 
where \(\mu \)  is introduced by (REF ).

Let \(\epsilon > 0\)  be a small number and let \(\theta =\pi \) . Let \(\Phi _0^{+}(x, 0)=\Phi _0^{\pi +\epsilon }(x, 0)\)  and
\(\Phi _0^{-}(x, 0)=\Phi _0^{\pi -\epsilon }(x, 0)\) 
be the actual fundamental matrix solutions at the origin of the DCHE built by Theorem REF . To find the Stokes matrix \(St_{\pi }\) 
we have to compare the solutions \(\Phi _0^{+}(x, 0)\)  and \(\Phi _0^{-}(x, 0)\)
\(\,\Phi _0^{-}(x, 0)=\Phi _0^{+}(x,0)\,\left(\begin{array}{cc}1 & \mu \\0 & 1\end{array}\right)\,.\,\) 
When \(\alpha \in {\mathbb {N}}\)  we find that
\(\mu &=&\frac{\gamma ^{\alpha -1}\,e^{-\frac{\beta }{x}}}{\beta }\,\left[\psi ^{-}_{\pi }(x) - \psi ^{+}_{\pi }(x)\right]=2\,\pi \,i\,\gamma ^{\alpha -1}\,e^{-\frac{\beta }{x}}\, {\mathrm {Res}} \left(\frac{v(\xi )\,e^{-\frac{\xi }{x}}}{\xi +\beta }\,;\,\xi =-\beta \right)\\[0.2ex]&=&2\,\pi \,i\,\gamma ^{\alpha -1}\,v(-\beta )=2\,\pi \,i\,\gamma ^{\alpha -1}\sum _{n=0}^{\infty } \frac{(-1)^n\,\beta ^n\,\gamma ^n}{n!\,\Gamma (\alpha +n)}\,.\) 
Similarly, when \(\alpha \in {\mathbb {Z}}_{\le 0}\)  we find that
\(\mu &=&\frac{x^{1-\alpha }\,e^{-\frac{\beta }{x}}}{\beta }\,\left[\phi ^{-}_{\pi }(x) - \phi ^{+}_{\pi }(x)\right]=2\,\pi \,i\,e^{-\frac{\beta }{x}}\, {\mathrm {Res}} \left(\frac{\xi ^{1-\alpha }\,q(\xi )\,e^{-\frac{\xi }{x}}}{\xi +\beta }\,;\,\xi =-\beta \right)\\[0.2ex]&=&2\,\pi \,i\,(-\beta )^{1-\alpha }\,q(-\beta )=2\,\pi \,i\,(-\beta )^{1-\alpha }\sum _{n=0}^{\infty } \frac{(-1)^n\,\beta ^n\,\gamma ^n}{n!\,\Gamma (2-\alpha +n)}\,.\) 
Now we will show that the so found multipliers coincide. Indeed, let \(\alpha \in {\mathbb {N}}\) . Then the multiplier
\(\mu \)  corresponding to \(\alpha \in {\mathbb {Z}}_{\le 0}\)  becomes
\(\,\mu =2 \,\pi \,i\,(-\beta )^{1-\alpha }\sum _{n=\alpha -1}^{\infty }\frac{(-1)^n\,\beta ^n\,\gamma ^n}{n!\,(1+n-\alpha )}\,\) 
since \(1/\Gamma (z)=0\)  for \(z\in {\mathbb {Z}}_{\le 0}\) . Then
\(\,\mu =2\,\pi \,i\,(-\beta )^{1-\alpha }\sum _{p=0}^{\infty }\frac{(-1)^{p+\alpha -1}\,\beta ^{p+\alpha -1}\,\gamma ^{p+\alpha -1}}{(p+\alpha -1)!\,p!}=2\,\pi \,i\,\gamma ^{\alpha -1}\sum _{p=0}^{\infty }\frac{(-1)^p\,\beta ^p\,\gamma ^p}{p!\,\Gamma (p+\alpha )}\,,\,\) 
which is the multiplier \(\mu \)  corresponding to \(\alpha \in {\mathbb {N}}\) .
Let now \(\alpha \notin {\mathbb {Z}}\)  and let us compare the functions \(\varphi ^{\pi -\epsilon }_k(x)\)  and \(\varphi ^{\pi +\epsilon }_k(x)\) . We
have
\(\,\varphi ^{\pi -\epsilon }_k(x) - \varphi ^{\pi +\epsilon }_k(x)=(\beta )^{2-\alpha +k}\int _{\gamma } (\beta +\xi )^{\varepsilon -k-2}\,e^{-\frac{\xi }{x}}\,d \xi \,,\,\) 
where \(\gamma =(\pi -\epsilon ) - (\pi +\epsilon )\) . Without changing the integral we can deform the path \(\gamma \)  into a Henkel
type path going along the negative real axis from \(-\infty \)  to \(-\beta \) , encircling \(-\beta \)  in the positive sense and
backing to \(-\infty \) . Then
\(\varphi ^{\pi -\epsilon }_k(x) - \varphi ^{\pi +\epsilon }_k(x) &=&\beta ^{2-\alpha +k}\,\left(1-e^{-2 \pi \,i(\alpha -k-2)}\right)\int _{-\beta }^{-\infty } (\beta +\xi )^{\alpha -k-2}\,e^{-\frac{\xi }{x}}\,d \xi \\[0.2ex]&=&\beta ^{2-\alpha +k}\, \left(1-e^{-2 \pi \,i\,\alpha }\right)\,e^{\frac{\beta }{x}}\int _0^{-\infty } u^{\alpha -k-2}\,e^{-\frac{u}{x}}\,d u\\[0.1ex]&=&\beta ^{2-\alpha +k}\,\left(1-e^{-2 \pi \,i\,\alpha }\right)\,e^{\frac{\beta }{x}}\,x^{\alpha -k-1}\,\int _0^{+\infty } \tau ^{\alpha -k-2}\,e^{-\tau }\,d \tau \\[0.15ex]&=&\beta ^{2-\alpha +k}\,\left(1-e^{-2 \pi \,i\,\alpha }\right)\,\Gamma (\alpha -k-1)\,x^{\alpha -k-1}\,e^{\frac{\beta }{x}}\\[0.15ex]&=&-\frac{2\,\pi \,i\,(-1)^k\,e^{-\pi \,i\,\alpha }\,\beta ^{2-\alpha +k}}{\Gamma (2-\alpha +k)}\,x^{\alpha -k-1}\,e^{\frac{\beta }{x}}\,,\) 
where we have used the Euler's reflection formula \(\Gamma (1-z)\,\Gamma (z)=\frac{\pi }{\sin \,\pi z}\)  fro \(z\notin {\mathbb {Z}}\) .
Then for the multiplier \(\mu \)  we find
\(\mu &=&\frac{x^{1-\alpha }\,e^{-\frac{\beta }{x}}}{\beta }\,\left[\varphi ^{-}_{\pi }(x) - \varphi ^{+}_{\pi }(x)\right]\\[0.2ex]&=&\frac{x^{1-\alpha }\,e^{-\frac{\beta }{x}}}{\beta }\,\sum _{k=0}^{\infty } \frac{\gamma ^k\,x^k}{k!}\,\left[\varphi ^{\pi -\epsilon }_k(x) - \varphi ^{\pi +\epsilon }_k(x)\right]\\[0.2ex]&=&2 \pi \,i\,(-\beta )^{1-\alpha }\sum _{k=0}^{\infty }\frac{(-1)^k\,\gamma ^k\,\beta ^k}{k!\,\Gamma (2-\alpha +k)}\,.\) 
This ends the proof.
\(\Box \)
Appendix
In this paragraph we will show by a direc computation that for lower values of \(\alpha \)
\(\,\lim _{\sqrt{\varepsilon } \rightarrow 0} (q_R + q_L)=\lambda \,\) 
where
\(\lambda =\frac{\gamma ^{\alpha -1}}{(\alpha -1)!}\,.\) 
Let \({\bf \alpha =2}\) . Then the multipliers \(q_R\)  and \(q_L\)  become
\(q_R=\frac{1}{2 \sqrt{\varepsilon }}\left(\frac{1+\varepsilon }{1-\varepsilon }\right)^{\frac{\gamma }{2 \sqrt{\varepsilon }}}\,\quad q_L=-\frac{1}{2 \sqrt{\varepsilon }}\left(\frac{1-\varepsilon }{1+\varepsilon }\right)^{\frac{\gamma }{2 \sqrt{\varepsilon }}}\,.\) 
Note that in this case \( q_R \rightarrow +\infty \)  and
\( q_L \rightarrow -\infty \)  when \(\sqrt{\varepsilon } \rightarrow 0\) .
Next the functions
\(\left(\frac{1-\varepsilon }{1+\varepsilon }\right)^{\frac{\gamma }{2 \sqrt{\varepsilon }}}\)  and
\(\left(\frac{1+\varepsilon }{1-\varepsilon }\right)^{\frac{\gamma }{2 \sqrt{\varepsilon }}}\)  are
expressed as power series in \(\sqrt{\varepsilon }\)  as follows
\(\left(\frac{1-\varepsilon }{1+\varepsilon }\right)^{\frac{\gamma }{2 \sqrt{\varepsilon }}} &=&\sum _{k=0}^{\infty }\left(-2 \gamma \sum _{p=o}^{\infty } \frac{(\sqrt{\varepsilon })^{4 p +1}}{4 p+2}\right)^k \frac{1}{k!}=1 - 2 \gamma \left(\frac{\sqrt{\varepsilon }}{2} + \frac{(\sqrt{\varepsilon })^5}{6} + \cdots \right)\nonumber \\[0.2ex]&+&\frac{4 \gamma ^2}{2 !}\left(\frac{\sqrt{\varepsilon }}{2} + \frac{(\sqrt{\varepsilon })^5}{6} + \cdots \right)^2 -\frac{8 \gamma ^3}{3!}\left(\frac{\sqrt{\varepsilon }}{2} + \frac{(\sqrt{\varepsilon })^5}{6} + \cdots \right)^3 + \cdots \\[0.2ex]\left(\frac{1+\varepsilon }{1-\varepsilon }\right)^{\frac{\gamma }{2 \sqrt{\varepsilon }}} &=&\sum _{k=0}^{\infty }\left(2 \gamma \sum _{p=o}^{\infty } \frac{(\sqrt{\varepsilon })^{4 p +1}}{4 p+2}\right)^k \frac{1}{k!}=1 + 2 \gamma \left(\frac{\sqrt{\varepsilon }}{2} + \frac{(\sqrt{\varepsilon })^5}{6} + \cdots \right)\nonumber \\[0.2ex]&+&\frac{4 \gamma ^2}{2 !}\left(\frac{\sqrt{\varepsilon }}{2} + \frac{(\sqrt{\varepsilon })^5}{6} + \cdots \right)^2 +\frac{8 \gamma ^3}{3!}\left(\frac{\sqrt{\varepsilon }}{2} + \frac{(\sqrt{\varepsilon })^5}{6} + \cdots \right)^3 + \cdots \nonumber \) 
Then we find that when \(\alpha =2\)
\(\lim _{\sqrt{\varepsilon } \rightarrow 0} (q_R+q_L)=\lim _{\sqrt{\varepsilon } \rightarrow 0}\frac{2 \gamma \,\sqrt{\varepsilon } + O (\varepsilon )}{2 \sqrt{\varepsilon }}=\gamma \,,\) 
which coincides with (REF ).
Let \({\bf \alpha =4}\) . Then the multipliers \(q_R\)  and \(q_L\)  become
\(q_R &=& -\left(\frac{1+\varepsilon }{1-\varepsilon }\right)^{\frac{\gamma }{2 \sqrt{\varepsilon }}}\frac{1}{(2 \sqrt{\varepsilon })^3}\left[\Gamma (3) - \Gamma (2)\,\frac{2 \sqrt{\varepsilon }\,\gamma }{1-\varepsilon ^2}\right],\\[0.2ex]q_L &=& \left(\frac{1-\varepsilon }{1+\varepsilon }\right)^{\frac{\gamma }{2 \sqrt{\varepsilon }}}\frac{1}{(2 \sqrt{\varepsilon })^3}\left[\Gamma (3) + \Gamma (2) \frac{ 2 \sqrt{\varepsilon }\,\gamma }{1-\varepsilon ^2}\right]\,.\) 
Note that in this case \(\lim _{\sqrt{\varepsilon } \rightarrow 0} q_R=-\infty \)  and
\(\lim _{\sqrt{\varepsilon } \rightarrow 0} q_L=+\infty \) . Again using the expressions (REF ) we find
that
\(\,\lim _{\sqrt{\varepsilon } \rightarrow 0} (q_R+q_L)=\frac{4}{3}\lim _{\sqrt{\varepsilon } \rightarrow 0}\frac{(\sqrt{\varepsilon })^3 \,\gamma ^3 + O (\varepsilon ^2)}{(2 \sqrt{\varepsilon })^3}=\frac{\gamma ^3}{3!}\,,\,\) 
which coincides with (REF ).
Let \({\bf \alpha =6}\) . Then the multipliers \(q_R\)  and \(q_L\)  become
\(q_R &=&\left(\frac{1+\varepsilon }{1-\varepsilon }\right)^{\frac{\gamma }{2 \sqrt{\varepsilon }}}\frac{1}{2!\,(2 \sqrt{\varepsilon })^5}\left[\frac{\Gamma (5)}{2!} -\Gamma (4)\,\frac{2\sqrt{\varepsilon }\,\gamma }{1-\varepsilon ^2}+\frac{\Gamma (3)}{2!}\,\left(\frac{2 \sqrt{\varepsilon }}{1-\varepsilon ^2}\right)^2 \right],\\[0.2ex]q_L &=&-\left(\frac{1-\varepsilon }{1+\varepsilon }\right)^{\frac{\gamma }{2 \sqrt{\varepsilon }}}\frac{1}{2!\,(2 \sqrt{\varepsilon })^5}\left[\frac{\Gamma (5)}{2!} +\Gamma (4)\,\frac{2\sqrt{\varepsilon }\,\gamma }{1-\varepsilon ^2}+\frac{\Gamma (3)}{2!}\,\left(\frac{2 \sqrt{\varepsilon }}{1-\varepsilon ^2}\right)^2 \right]\,.\) 
Then \(\lim _{\sqrt{\varepsilon } \rightarrow 0} q_R=+\infty \)  while \(\lim _{\sqrt{\varepsilon } \rightarrow 0} q_L=-\infty \) .
For the sum \(q_R + q_L\)  we have
\(\,\lim _{\sqrt{\varepsilon } \rightarrow 0} (q_R + q_L)=\frac{4}{15} \lim _{\sqrt{\varepsilon } \rightarrow 0}\frac{(\sqrt{\varepsilon })^5\,\gamma ^5 + O (\varepsilon )^3}{(2 \sqrt{\varepsilon })^5}=\frac{\gamma ^5}{5!}\,,\,\) 
which coincides with (REF ).
